{"id":"n-632e251b82cedece","revision":0,"first_index":1,"fingerprint":"f6-2fb59f4c2ac3af18","name":"theta_plus_one of Triangle read by rows: T(n,k) = n!*(n+k-1)!/((n-k)!*(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","2"],["1","8","9"],["1","18","54","40"],["1","32","180","320","175"],["1","50","450","1400","1750","756"],["1","72","945","4480","9450","9072","3234"],["1","98","1764","11760","36750","58212","45276","13728"],["1","128","3024","26880","115500","266112","336336","219648","57915"],["1","162","4860","55440","311850","972972","1765764","1853280","1042470","243100"],["1","200","7425","105600","750750","3027024","7357350","10982400","9845550","4862000","1016158"],["1","242","10890","188760","1651650","8324316","25897872","51342720","64980630","50807900","22355476","4232592"],["1","288","15444","320320","3378375","20756736","80047968","201636864","336717810","369512000","256071816","101582208","17577014"],["1","338","21294","520520","6506500","47783736","222990768","691726464","1459110510","2101599500","2034348316","1265545008","457002364","72804200"],["1","392","28665","815360","11911900","102918816","570341772","2128389120","5499724230","9958348400","12597310726","10903156992","6151954900","2038517600","300874500"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","18","113","708","4407","27254","167589","1025544","6249899","37954458","229793049","1387616844","8359959519","50264293374"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1","2"],["1","8","9"],["1","18","54","40"]],"recurrence":{"denominator":{"coeffs":[["126"],["-253"],["167"],["-44"],["4"]]},"terms":[{"coeff":{"coeffs":[["-300"],["455"],["-239"],["52"],["-4"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["1056","2160"],["-1640","-3324"],["888","1784"],["-200","-400"],["16","32"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-1350","-3672","-3780"],["2172","5856","5952"],["-1226","-3280","-3296"],["288","768","768"],["-24","-64","-64"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["720","1512"],["-1240","-2532"],["744","1496"],["-184","-368"],["16","32"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177298,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-dd0f94e7f4596445"]},{"id":177299,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-dd0f94e7f4596445"]},{"id":177300,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-dd0f94e7f4596445"]}],"region_exact":{},"incoming_edges":[{"id":"e-dd0f94e7f4596445","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-632e251b82cedece","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"3c2d1285f7c144ec10ff8367c48fc44ddf7168058b3a1eaaaeab37e7b359cd75","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-55697ddae4d22d60","role":"sequence","name":"Triangle read by rows: T(n,k) = n!*(n+k-1)!/((n-k)!*(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,   2;\n   3: 1,   8,    9;\n   4: 1,  18,   54,    40;\n   5: 1,  32,  180,   320,    175;\n   6: 1,  50,  450,  1400,   1750,    756;\n   7: 1,  72,  945,  4480,   9450,   9072,    3234;\n   8: 1,  98, 1764, 11760,  36750,  58212,   45276,   13728;\n   9: 1, 128, 3024, 26880, 115500, 266112,  336336,  219648,   57915;\n  10: 1, 162, 4860, 55440, 311850, 972972, 1765764, 1853280, 1042470, 243100;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4-44*n^3+167*n^2-253*n+126)*P(n,x) = -(4*n^4-52*n^3+239*n^2-455*n+300)*P(n-4,x) + (16*n^4-200*n^3+888*n^2-1640*n+1056+(32*n^4-400*n^3+1784*n^2-3324*n+2160)*x)*P(n-3,x) - (24*n^4-288*n^3+1226*n^2-2172*n+1350+(64*n^4-768*n^3+3280*n^2-5856*n+3672)*x+(64*n^4-768*n^3+3296*n^2-5952*n+3780)*x^2)*P(n-2,x) + (16*n^4-184*n^3+744*n^2-1240*n+720+(32*n^4-368*n^3+1496*n^2-2532*n+1512)*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + 2*x, P(3,x) = 1 + 8*x + 9*x^2, P(4,x) = 1 + 18*x + 54*x^2 + 40*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 2*x;\nP[3, x_] := 1 + 8*x + 9*x^2;\nP[4, x_] := 1 + 18*x + 54*x^2 + 40*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*P[n - 4, x] + (16*n^4 - 200*n^3 + 888*n^2 - 1640*n + 1056 + (32*n^4 - 400*n^3 + 1784*n^2 - 3324*n + 2160)*x)*P[n - 3, x] - (24*n^4 - 288*n^3 + 1226*n^2 - 2172*n + 1350 + (64*n^4 - 768*n^3 + 3280*n^2 - 5856*n + 3672)*x + (64*n^4 - 768*n^3 + 3296*n^2 - 5952*n + 3780)*x^2)*P[n - 2, x] + (16*n^4 - 184*n^3 + 744*n^2 - 1240*n + 720 + (32*n^4 - 368*n^3 + 1496*n^2 - 2532*n + 1512)*x)*P[n - 1, x])/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 1 + 8*x + 9*x^2; v[4] = 1 + 18*x + 54*x^2 + 40*x^3; for(n = 5, N, my(i = n); v[i] = (-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*v[i - 4] + (16*n^4 - 200*n^3 + 888*n^2 - 1640*n + 1056 + (32*n^4 - 400*n^3 + 1784*n^2 - 3324*n + 2160)*x)*v[i - 3] - (24*n^4 - 288*n^3 + 1226*n^2 - 2172*n + 1350 + (64*n^4 - 768*n^3 + 3280*n^2 - 5856*n + 3672)*x + (64*n^4 - 768*n^3 + 3296*n^2 - 5952*n + 3780)*x^2)*v[i - 2] + (16*n^4 - 184*n^3 + 744*n^2 - 1240*n + 720 + (32*n^4 - 368*n^3 + 1496*n^2 - 2532*n + 1512)*x)*v[i - 1])/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-55697ddae4d22d60","steps":["\u03b8+1"],"edges":["e-dd0f94e7f4596445"],"text":"A123160 \u2192 \u03b8+1"},"headline":"A123160 \u2192 \u03b8+1","chain":null}